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Birthday Probabilities

Tool to calculate the birthday paradox problem. The birthday problem is famous in probabilities because its results are non-intuitive. It allows answering how many people are necessary to have 50% chance that 2 of them share the same birthday.

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Birthday Probabilities -

Tag(s) : Statistics, Fun/Miscellaneous

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# Birthday Probabilities

### Probability to share a same birthday

 Probability That at least N people that exactly N people That not any one
 To be born a same day (at any date) a given date (ie. May 1st)

Tool to calculate the birthday paradox problem. The birthday problem is famous in probabilities because its results are non-intuitive. It allows answering how many people are necessary to have 50% chance that 2 of them share the same birthday.

### What is the birthday paradox ? (Definition)

The paradox of birthdays is a mathematical problem put forward by Von Mises, who looks for the value N in the problem: In a group of N people there is 50% chance that at least 2 people in the group share the same birthday (day + month). The answer is N = 23, which is quite counterintuitive, hence the paradox.

### What are the hypothesis made to calculate the birthday probabilities?

During the calculation of the birthdate paradox, it is supposed that births are equally distributed over the days of a year (it is not exactly true in reality). In the following, a year has 365 days (leap years are ignored).

### What is the probability for a person to be born a given day of the year?

Probability is 1/365 = 0.0027 = 0.27%, indeed, one chance out of 365 to be bord a precise day.

### What is the probability for a person to be born a different day of mine?

The probability is 364/365 = 0.9973 = 99.73%, indeed, he must be born a distinct day of mine, so there is 364 possibilities out of 365.

### What is the probability for a person to be born the same day as me?

This calculation is the same as What is the probability for a person to be born a given day of the year? This given day is my birthday.

Example: 1/365 = 0.0027 = 0.27%

The calculation could also be formulated by noting that the probability for a person to be born the same day as me is the opposite as the probability for a person to be born a different day than mine.

Example: 1 - 364/365 = 1 - 0.9973 = 0,0027 = 0.27%

### What is the probability for a mother and its child to be born the same day?

This calculation is the same as What is the probability for a person (the child) to be born a given day of the year (the mother birthday)?.

Example: 1/365 = 0.0027 = 0.27%

### What are the odds for 2 people in a group of N to be born the same day?

For N = 2, it is the same as probability for a person to be born the same day as me or the opposite as the probability for a person to be born a different day than mine.

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

For other people, they have to be born also a distinct day than me, but also a distinct day of the other one (then calculate the opposite):

Example: P(N=3) = 1 - (364/365) * (363/365) = 0,0082 = 0.82%

Example: P(N=4) = 1 - (364/365) * (363/365) * (362/365) = 0,0164 = 1.64%

Example: P(N=5) = 1 - (364/365) * (363/365) * (362/365) * (361/365) = 0,0271 = 2.71%

### What is the probability that among 23 people, 2 share the same birth day?

About 0.5, 50% (1 chance out of 2)

### How many people are needed in a group to be sure that 2 share the same birthday ?

366 are needed (367 if leap years are taken into account). Indeed, every day of the year plus one are needed to be sure that at least a couple of two people share the same birthday.

### How many people are necessary to have the probability > 50% of 2 people to be born the same given day ?

For a given date, 253 people are needed to get a probability of 50% that 2 people to be born a same precise day.

### What is the probability of 2 people among N to be born a given day?

Probabilities are multiplied

Example: P(N=1) = 1 - (364/365)^(N-1).

Example: P(N=2) = 1 - (364/365) = 0,0027 = 0.27%

Example: P(N=3) = 1 - (364/365)^2 = 0,0055 = 0.55%

Example: P(N=n) = 1 - (364/365)^(n-1)

### What is the probability that N people not to be born a given day of the year?

The probability is

Example: P = (364/365)^N

### How many people are born the same day as myself?

With the hypothesis of a world population of 8 billion people, there are 8000000000 / 365.25 or about 22 million people born on a given day / month.

Limiting to USA (350 millions inhabitants), there are 350000000 / 365.25 or nearly 1 millions US people born on a given day / month date.

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